We have measured the spectral energy distribution ( SED ) of the host galaxy of the z _ { s } = 1.7 gravitationally lensed quasar SDSS J1004+4112 from 0.44 - 8.0 \mu m ( 0.16 - 3.0 \mu m in the rest frame ) .
The large angular extent of the lensed images and their separation from the central galaxy of this cluster lens allows the images to be resolved even with the Spitzer Space Telescope .
Based on the SED , the host galaxy is a mixture of relatively old and intermediate age stars with an inferred stellar mass of \log ( M _ { \star } / M _ { \sun } ) = 11.09 \pm 0.28 and a star formation rate of \log ( \dot { M } / M _ { \odot } yr ^ { -1 } ) = 1.21 \pm 0.26 .
Given the estimated black hole mass of M _ { BH } \simeq 10 ^ { 8.6 } M _ { \odot } from locally-calibrated correlations of black hole masses with line widths and luminosities , the black hole represents a fraction \log ( M _ { BH } / M _ { \star } ) = -2.49 \pm 0.28 of the stellar mass and it is radiating at 0.24 \pm 0.05 of the Eddington limit .
The ratio of the host stellar mass to the black hole mass is only marginally consistent with the locally observed ratio .